I was channel flipping yesterday and ta-da, here’s an Space Shuttle Astronaut (don’t know his name) going for the Million dollar question…
The question was: “How many Common Factors do the numbers 28 and 32 have?”
He said two (2 and 4) and the answer was three (1, 2, and 4).
Is that right? I always thought that the number one was not included for Common Factors. For example, relative prime numbers (such as 4 and 15) have no Common Factors by definition. According to Fox, they always will have at least one.
When you factor something to prime factors, you leave out one (because it’s not generally considered prime, and everything would have it in any case), which I suspect is what confused the contestant, as well.
But they just asked for common factors, period, and one is certainly a common factor.
This is an illustration of the failing of multiple-choice tests. Either answer (1, 2, and 4 or 2 and 4) makes it clear that the person understands the problem and its solution, but on a multiple choice test, one of them is going to be marked wrong.
Given two natural numbers, we can ask for their greatest common factor (aka “greatest common divisor” or GCD). If the numbers are relatively prime, their greatest common factor is 1. If 1 were not considered a common factor, then a question like “What’s the GCD of 8 and 15?” wouldn’t have an answer.
God, is this show still on? The few times I saw it a couple years ago almost all of the questions were stupidly easy, except the million dollar question wasn’t ever worth the risk. You either walked away with 500 grand, or risked it all before seeing a question the avg 5th grader would have no idea about. The one I saw was about the name of one of the space shuttles, and the answer was the Eagle III or something.
Please forgive my ignorance. I took as few math classes as humanly possible and still graduate. What is a relative prime number? And how is 4 any type of prime number? Thanks.
In case Giles’s explanation didn’t suffice, being relatively prime isn’t a property of individual numbers; it’s a relationship between numbers. It’s kinda like if two people don’t have any mutual friends. (So, there’s no such thing as a relatively prime number.)
Perhaps an example will help. The number 4 and 15 are relatively prime, althogh neither is a prime number, because:
The factors of 4 are 1, 2 and 4.
The factors of 15 are 1, 3, 5 and 15.
So the only common factor, and hence the greatest common factor, is 1.
